Question

Simplify the expression. \(3^{\log _3 5 x}\)

Short answer

The simplified form of the expression \(3^{\log _3 5x}\) is \(5x\).

Step-by-step solution

Apply the Base-Exponent Logarithmic Property

The expression given is \(3^{\log _3 5x}\). From the properties of logarithms, it is known that for any number a and b, where a is greater than 0 and a is not equal to 1, the expression \(a^{\log_a b}\) simplifies to b. Here, a is 3, and b is \(5x\).

Simplify the Expression

By applying the property discussed in the previous step, we can simplify the given expression. This changes the expression \(3^{\log _3 5x}\) to simply \(5x\).

Key concepts

Base-Exponent Logarithmic Property

One of the fundamental concepts in working with logarithms in Algebra 2 is the Base-Exponent Logarithmic Property. This powerful property allows you to simplify expressions in a straightforward way. If you have an expression of the form \(a^{\log_a b}\), it simplifies directly to \(b\). Why does this work, you might wonder? Well, think of the logarithm \(\log_a b\) as the power you raise \(a\) to get \(b\). Therefore, raising \(a\) to its logarithmic power returns you to the original base of \(b\).
Here’s how you can visualize it:

• The base \(a\) of both the exponent and the logarithm match perfectly.
• The exponent—\(\log_a b\)—is essentially the power needed to transform \(a\) to \(b\).
• So, \(a^{\log_a b} = b\) simplifies directly to \(b\).

Using this property saves time and effort while simplifying. Remember, the bases must match for the property to apply!

Simplifying Expressions

Simplifying an expression like \(3^{\log_3 5x}\) using properties of logarithms can make tackling Algebra problems much simpler. It's all about recognizing when and how these properties can be applied.
In our example, see how the immediate recognition of the Base-Exponent Logarithmic Property leads to simplification. Instead of manually expanding or attempting to calculate complex values, we use direct properties to make the expression simpler and easier to handle. Here are some tips:

• Check if the base of the exponent matches that of the logarithm.
• If they do, remember that the expression can be simplified as \(a^{\log_a b} = b\).
• Apply this knowledge to turn lengthy expressions into simple ones quickly.

Recognizing opportunities to apply such properties is key in simplifying expressions and overcoming complex algebraic problems.

Algebra 2 Concepts

Algebra 2 often involves deeper exploration of mathematical properties, like logarithms, to simplify equations and expressions. Logarithms themselves are simply another way of expressing exponents. In Algebra 2, you learn to manipulate these logarithmic expressions efficiently to solve problems.
Some core areas you'll delve into include:

• Understanding and applying logarithmic properties such as the product, quotient, and power rules.
• Solving exponential equations using logarithms.
• Graphing logarithmic functions to understand their behavior and transformations.

These skills build up your mathematical toolkit, enabling you to tackle a wide range of problems. The simplification of expressions using properties, as demonstrated in our example, is just one way that Algebra 2 concepts are applied. By embracing these concepts, you're setting up a solid foundation for further studies in mathematics.